Semidefinite programming (SDP) solvers are increasingly used as primitives in many program verification tasks to synthesize and verify polynomial invariants for a variety of systems including programs, hybrid systems and stochastic models. On one hand, they provide a tractable alternative to reasoning about semi-algebraic constraints. However, the results are often unreliable due to “numerical issues” that include a large number of reasons such as floating-point errors, ill-conditioned problems, failure of strict feasibility, and more generally, the specifics of the algorithms used to solve SDPs. These issues influence whether the final numerical results are trustworthy or not. In this paper, we briefly survey the emerging use of SDP solvers in the static analysis community. We report on the perils of using SDP solvers for common invariant synthesis tasks, characterizing the common failures that can lead to unreliable answers. Next, we demonstrate existing tools for guaranteed semidefinite programming that often prove inadequate to our needs. Finally, we present a solution for verified semidefinite programming that can be used to check the reliability of the solution output by the solver and a padding procedure that can check the presence of a feasible nearby solution to the one output by the solver. We report on some successful preliminary experiments involving our padding procedure.

中文翻译：

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验证多项式不变量的数值半定规划求解器

半定规划 (SDP) 求解器越来越多地用作许多程序验证任务中的原语，以合成和验证各种系统的多项式不变量，包括程序、混合系统和随机模型。一方面，它们为有关半代数约束的推理提供了一种易于处理的替代方法。然而，由于“数值问题”，结果往往不可靠，其中包括大量原因，例如浮点错误、病态问题、严格可行性失败，以及更一般地，用于解决 SDP 的算法的细节. 这些问题会影响最终的数值结果是否可信。在本文中，我们简要调查了静态分析社区中 SDP 求解器的新兴用途。我们报告了将 SDP 求解器用于常见的不变综合任务的危险，描述了可能导致不可靠答案的常见故障。接下来，我们演示了用于保证半定规划的现有工具，这些工具通常证明不足以满足我们的需求。最后，我们提出了一种验证半定规划的解决方案，可用于检查求解器输出的解决方案的可靠性，以及一种填充程序，可以检查求解器输出的可行邻近解是否存在。我们报告了一些涉及我们的填充程序的成功初步实验。我们提出了一种经过验证的半定规划的解决方案，可用于检查求解器输出的解决方案的可靠性，以及一种填充程序，可以检查求解器输出是否存在可行的邻近解。我们报告了一些涉及我们的填充程序的成功初步实验。我们提出了一种经过验证的半定规划的解决方案，可用于检查求解器输出的解决方案的可靠性，以及一种填充程序，可以检查求解器输出是否存在可行的邻近解。我们报告了一些涉及我们的填充程序的成功初步实验。